Direct product of A5 and Z2

This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this group
View a complete list of particular groups (this is a very huge list!)[SHOW MORE]

VIEW FACTS ABOUT THIS GROUP: All factsGroup property satisfactionsSubgroup property satisfactionsGroup property dissatisfactionsSubgroup property satisfactions
VIEW DEFINITIONS USING THIS AS EXAMPLE: All terms |Group properties |Subgroup properties|
VIEW FACTS USING THIS AS EXAMPLE: All facts |Group property non-implications |Subgroup property non-implications |Group metaproperty dissatisfactionsSubgroup metaproperty dissatisfactions

Definition

This group is defined in the following equivalent ways:

1. It is the full icosahedral group: it is the group of all rigid symetries of the regular icosahedron, including both orientation-preserving symmetries and orientation-reversing symmetries.
2. It is the external direct product of the alternating group of degree five and the cyclic group of order two.

Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 120#Arithmetic functions

Basic arithmetic functions

Function	Value	Similar groups	Explanation for function value
order (number of elements, equivalently, cardinality or size of underlying set)	120	groups with same order	order of direct product is product of orders, so the order is $|A_5||\mathbb{Z}_2| = (5!/2)(2) = 5! = 120$
exponent of a group	30	groups with same order and exponent of a group | groups with same exponent of a group	exponent of direct product is lcm of exponents, so lcm of exponents of $A_5$ and $\mathbb{Z}_2$, which is $\operatorname{lcm} \{ 30, 2 \} =30$
composition length	2	groups with same order and composition length | groups with same composition length	direct product of two simple groups
chief length	2	groups with same order and chief length | groups with same chief length	direct product of two simple groups
max-length	5	groups with same order and max-length | groups with same max-length
Frattini length	1	groups with same order and Frattini length | groups with same Frattini length
nilpotency class	not a nilpotent group
derived length	not a solvable group
Fitting length	not a solvable group

Group properties

Property	Satisfied?	Explanation
abelian group	No
nilpotent group	No
solvable group	No
simple group, simple non-abelian group	No
almost simple group	No
quasisimple group	No
almost quasisimple group	No
semisimple group	No
perfect group	No
directly indecomposable group	No

GAP implementation

Group ID

This finite group has order 120 and has ID 35 among the groups of order 120 in GAP's SmallGroup library. For context, there are 47 groups of order 120. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(120,35)

For instance, we can use the following assignment in GAP to create the group and name it $G$:

gap> G := SmallGroup(120,35);

Conversely, to check whether a given group $G$ is in fact the group we want, we can use GAP's IdGroup function:

IdGroup(G) = [120,35]

or just do:

IdGroup(G)

to have GAP output the group ID, that we can then compare to what we want.

Other descriptions

Description	Functions used
DirectProduct(AlternatingGroup(5),CyclicGroup(2))	DirectProduct, AlternatingGroup, CyclicGroup